Inclusive modeling and digital compensation for coupling impairments of high-speed coupled electrical microstrip traces in data center connections

ABSTRACT

Disclosed is a method of compensating for coupling impairments in optical communication systems comprising two or more coupled transmission lines. The method comprises introducing into the system a component comprising an electronic circuit configured to realize at least one of an exact electrical coupling compensation (EECC) algorithm and a spectrally fragmented electrical coupling compensation (SF-ECC) algorithm.

FIELD OF THE INVENTION

The invention is from the field of systems comprising coupledtransmission lines. Specifically the invention is related to the fieldof coupled transmission lines that experience coupling impairments thatrequire compensation.

BACKGROUND OF THE INVENTION

Publications and other reference materials referred to herein arenumerically referenced in the following text and respectively grouped inthe appended Bibliography which immediately precedes the claims.

The increased data capacity in cloud servers requires reliabletransmission at the highest available data rate. Therefore, opticalcommunication between the server units is used due its high data ratetransmission capabilities. In order to fit the aggressive low costrequirements of data centers, the next generation transceivers are basedon commercially available 25G direct detection components combined withstate-of-the-art communication and digital signal processing (DSP)techniques. Currently, 100G data center transceivers include a hybridoptical-electrical module packaged in a single pluggable form-factor.However, the hybrid transceiver poses system design challenges forperformance optimization, package size, and power dissipation. Toovercome these challenges, the next generation transceiver may includeonly the optical components, while the electronic blocks will be hostedexternally on the transponder board. This separation enables the use ofsmall form-factor pluggable (SFP) packages, where the aggregated opticalsignal consists of up to four independent channels, each carrying dataat a rate of 25 Gbaud and higher. In turn, each of the optical channelsis detected and converted to an electrical signal by the optical module.The electrical signal is transmitted over a printed transmission line(PTL) from the optical module to the electronic module. Both modules arehosted on the host printed circuit board (PCB), e.g., backplane.

Effective area utilization of a PCB is very important forcost-effective, compact and high port density solutions. Therefore, foreffective use of the PCB area, the transmission lines (TLs) and thecomponents should be integrated tightly. In next generation data centertransceivers, each of the PTLs is expected to support 25 Gbaud andhigher, i.e., ultra-broadband electrical signals. However, denseintegration of the TLs in high-speed applications can intensifyimpairments, such as crosstalk, which in turn leads to signaldegradation. The common techniques to reduce those impairments includeproperly separated and well-shielded TLs, or multilayer design withisolated high-speed lines. However, those techniques are tailor-made fora specific application and increase the cost and the required area ofthe PCB compared to low-speed applications where those impairments arenegligible.

Coupled PTLs have been studied in various RF and microwave applicationsand accordingly compensation techniques have been proposed. Mbairi etal. studied the crosstalk between adjacent high-frequency printedtransmission lines (TLs) as a function of traces separation [1]. It wasshown that the crosstalk for coupled microstrips and coupled striplinevaried considerably with frequency and should not be ignored. It wasproposed to increase the adjacent traces separation, but this reducesthe effective utilization of the PCB. On the other hand, Prachumrasee etal. studied the crosstalk between coupled microstrips in hard driveapplications [2]. The crosstalk has been suppressed by usingdifferential lines and a magnetic composite. But, the differential linesrequired more area compared to single-ended lines [3]. Alternatively,Pelard et al. present integrated circuit solutions that compensate forcrosstalk and intersymbol interference (ISI) of high-speed datatransmission over legacy systems, e.g., short reach optics andelectrical backplanes [4]. They showed that crosstalk canceller (XTC)improves the bit error rote (BER) significantly and enables 10 Gbps datarates on legacy systems. Also, in order to mitigate the crosstalkimpairment in coupled microstrips, Kao et al. presented a 10 Gbpsparallel receiver with joint XTC and decision-feedback equalizer (DFE)[5]. It was demonstrated that the adaptive receiver can compensate forchannel loss and cancel the far end crosstalk (FEXT) simultaneously.Similarly, Oh et al. designed a multiple high-speed I/Os XTC analogfront-end that handles the crosstalk of coupled microstrips at 12 Gbpsand improved the eye opening considerably [3]. Recently, Han et al.realized a 2×50 Gbps receiver with adaptive channel loss equalizer andXTC using a SiGe BiCMOS process [6]. The adaptive joint XTC equalizercompensated for the crosstalk and loss of the coupled microstrips withcapacitive- or inductive-coupling nature. However, all the electricalanalog XTC techniques in [3-6] are limited by their compensation, suchthat wideband signals with pronounced coupling and significantmicrostrip length may require a more accurate compensation model.

It is therefore a purpose of the present invention to provide inclusivemethods of compensating for coupling impairments that occur in systemscomprising two or more transmission lines.

It is another purpose of the present invention to provide inclusivecompensation methods for the coupling impairments of coupled-pairmicrostrips in optical communication systems that improve the receiversensitivity and support significantly longer microstrip traces ascompared to the classical crosstalk compensation technique.

Further purposes and advantages of this invention will appear as thedescription proceeds.

SUMMARY OF THE INVENTION

In a first aspect the invention is a method of compensating for couplingimpairments in transmission systems comprising two or more coupledtransmission lines. The method comprises introducing into the system acomponent comprising an electronic circuit that is configured to realizeat least one of: (a) an exact electrical coupling impairmentscompensation (EECC) algorithm and (b) a spectrally fragmented electricalcoupling compensation (SF-ECC) algorithm, wherein the SF-ECC can be usedfor compensation of coupling impairments In the frequency domain by afrequency-domain SF-ECC algorithm, and in the time domain by atime-domain SF-ECC algorithm.

In embodiments of the method the system is an ultra-broadbandoptoelectronic system comprising coupled pair-microstrip (CP-MS) traces.

In embodiments of the method the EECC algorithm comprises:

${\underset{\_}{v}}_{n}^{({EECC})} = {{IDTFT}\{ {{{{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} \rbrack}_{{t = \frac{n \cdot T_{s}}{SPS}}}} \} \}} }$wherein v _(n) ^((EECC)) is the compensated digital signal vector, n isthe sampling index, DTFT stands for the discrete-time Fourier transform,IDTFT is the inverse DTFT, V_(in)(t,0) is the time-domain injectedultra-broadband electrical signal column vector at the CP-MS inputs,h_(ms)(t) is the coupling impulse response matrix ℑ⁻¹{H_(MS) (jω)}, ℑstands for the Fourier transform, T_(s) is the sampling period, SPS isthe number of samples-per-symbol, * denotes the convolution operation,Ĥ_(MS) ⁻¹(jω′) is the inverse matrix of the sampled coupling transferfunction, ω′∈[−ω_(s)/2, ω_(s)/2] is the digital angular frequency, andω_(s) is the angular sampling frequency that follows the samplingfrequency f_(s) of the analog to digital (ADC), which is related to thesampling period by T_(s)=2π/ω_(s). In these embodiments for CP-MS tracesthe inverse matrix of the sampled coupling transfer function is:

${{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} = {\begin{bmatrix}{\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )} & {j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} \\{j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} & {\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}\end{bmatrix}.}$

In embodiments of the method the SF-ECC algorithm is:

${\underset{\_}{v}}_{n}^{({{SF} - {EECC}})} = {{IDTFT}\{ {{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} \cdot {DTFT}}{\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} \rbrack}_{{t = \frac{n \cdot T_{s}}{SPS}}}} \} \}.}} }$

In these embodiments the frequency-domain SF-ECC algorithm comprises:

$\begin{matrix}{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} = \begin{bmatrix}{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} & {- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} \\{- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} & {{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )}\end{bmatrix}} & \; \\{{where},} & \; \\{{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} = {\cos\mspace{11mu}( {\omega{\sum\limits_{k = 1}^{M}{{\Delta\tau}_{MS}^{(k)}(\omega)}}} )}} & \; \\{and} & \; \\{{{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} = {{- j}\mspace{11mu}\sin\mspace{11mu}( {\omega{\sum\limits_{k = 1}^{M}{{\Delta\tau}_{MS}^{(k)}(\omega)}}} )}} & \;\end{matrix}$and the compensated digital signal vector is:

$\mspace{79mu}{{\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {{IDTFT}{\{ {{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)} \star {v_{\ln}( {t,0} )}} \rbrack_{{|t} = \frac{n \cdot T_{s}}{SPS}} \}} \}.}}}$

In these embodiments the time-domain SF-ECC algorithm comprises:

$\begin{matrix}{\mspace{79mu}{{{\overset{\sim}{h}}_{MS}^{- t}(t)} = \begin{bmatrix}{{\overset{\sim}{h}}_{IL}^{({MS})}(t)} & {- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} \\{- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} & {{\overset{\sim}{h}}_{IL}^{({MS})}(t)}\end{bmatrix}}} & \; \\{\mspace{79mu}{where}} & \; \\{\mspace{79mu}{\begin{matrix}{{{\overset{\sim}{h}}_{IL}^{({MS})}(t)} = {\mathcal{J}^{- 1}\{ {{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} \}}} \\{= {\sum\limits_{k = 1}^{M}{{\overset{\sim}{h}}_{IL}^{{({MS})}{(k)}}(t)}}}\end{matrix},}} & \; \\{\mspace{79mu}{\begin{matrix}{{{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)} = {\mathcal{J}^{- 1}\{ {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} \}}} \\{= {\sum\limits_{k = 1}^{M}{{\overset{\sim}{h}}_{FEXT}^{{({MS})}{(k)}}(t)}}}\end{matrix},}} & \; \\{\mspace{79mu}\begin{matrix}{{{\overset{\sim}{h}}_{IL}^{{({MS})}{(k)}}(t)} = {\mathcal{J}^{- 1}\{ {{\overset{\sim}{H}}_{IL}^{{({MS})}{(k)}}( {j\;\omega} )} \}}} \\{= {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{\cos( {{\omega\Delta\tau}_{MS}(\omega)} )}{\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}d\;\omega}}}} \\{\cong {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{\cos( {{\omega\Delta\tau}_{{MS}_{(1)}}^{(k)}(\omega)} )}{\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}d\;\omega}}}} \\{= {{\frac{1}{4\pi}{\int_{- \infty}^{\infty}{e^{{{j\;{\omega{({{\Delta\tau}_{{MS}_{0}}^{(k)} - {\omega_{0}^{(k)}{\Delta\tau}_{{MS}_{1}}^{(k)}}})}}} + {j\;\omega^{2}{\Delta\tau}_{{MS}\; 1}^{(k)}}}\;}{\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}d\;\omega}}} +}} \\{= {\frac{1}{4\pi}{\int_{- \infty}^{\infty}{e^{{{- j}\;{\omega{({{\Delta\tau}_{{MS}_{0}}^{(k)} - {\omega_{0}^{(k)}{\Delta\tau}_{{MS}_{1}}^{(k)}}})}}} - {j\;\omega^{2}{\Delta\tau}_{{MS}_{1}}^{(k)}}}\;{\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}d\;\omega}}}} \\{{= {{A\begin{bmatrix}{{{\sqrt{j}e^{- {\vartheta^{(k)}{(t)}}}} \star {\delta( {t + \gamma^{(k)}} )}} +} \\{{\sqrt{- j}e^{j\;{\vartheta^{(k)}{(t)}}}} \star {\delta( {t - \gamma^{(k)}} )}}\end{bmatrix}} \star {{{sinc}( {\frac{B_{M}}{2}t} )}e^{j\;\omega_{0}^{(k)}t}}}}\;}\end{matrix}} & \; \\{\mspace{79mu}{and}} & \; \\{\mspace{79mu}{{{\overset{\sim}{h}}_{FEXT}^{{({MS})}{(k)}}(t)} \cong {{A \cdot \begin{bmatrix}{{{{- \sqrt{j}}e^{{- j}\;{\vartheta^{(k)}{(t)}}}} \star {\delta( {t + \gamma^{(k)}} )}} +} \\{{\sqrt{- j}e^{j\;{\vartheta^{(k)}{(t)}}}} \star {\delta( {t - \gamma^{(k)}} )}}\end{bmatrix}} \star {{{sinc}( {\frac{B_{M}}{2}t} )}e^{j\;\omega_{0}^{(k)}t}}}}} & \;\end{matrix}$and the compensated digital signal vector is:

$\mspace{79mu}{{\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {\{ {{{\overset{\sim}{h}}_{MS}^{- 1}(t)} \star \{ \lbrack {{h_{MS}(t)}^{*}{v_{in}( {t,0} )}} \rbrack_{{|t} = \frac{n \cdot T_{s}}{SPS}} \}} \}.}}$

In embodiments of the method the EECC algorithm can be utilized forcoupled multiple transmission lines by using the exact inverse matrixsampled coupling transfer function of the coupled multiple transmissionlines.

In embodiments of the method the SF-ECC algorithm can be utilized forcoupled multiple transmission lines by using the approximatedfrequency-domain transfer function of the coupled multiple transmissionlines or by using the approximated time-domain impulse response ofcoupled multiple transmission lines.

In embodiments of the method the component that comprises an electroniccircuit configured to realize at least one of: the EECC algorithm, thefrequency-domain SF-ECC algorithm, and the time-domain SF-ECC algorithmis a digital signal processing (DSP) unit. Embodiments of the DSP unitcan be extended to compensate for intersymbol interference (ISI) byaddition of an interpolator and a feed-forward equalizer and decisionfeedback equalizer (FFE-DFE).

The DSP unit can be located at one of:

-   -   a) the transmitter side of the optoelectronic system for        pre-compensation implementation of the coupling impairments by        the EECC or SF-ECC algorithms; and    -   b) the receiver side of the optoelectronic system for        post-compensation implementation of the coupling impairments by        the EECC or SF-ECC algorithms.

In embodiments of the method one DSP unit is located at the transmitterside of the optoelectronic system for pre-compensation implementation ofthe coupling impairments by the EECC or SF-ECC algorithms and a secondDSP unit comprising an interpolator and a FFE-DFE is located at thereceiver side of the optoelectronic system.

In a second aspect the invention is a digital signal processing (DSP)unit for an ultra-broadband optoelectronic system. The DSP unitcomprises an electronic circuit configured to realize at least one of anEECC algorithm, a frequency-domain SF-ECC algorithm, and a time-domainSF-ECC algorithm.

Embodiments of the DSP unit additionally comprise an interpolator and afeed-forward equalizer and decision feedback equalizer (FFE-DFE).

All the above and other characteristics and advantages of the inventionwill be further understood through the following illustrative andnon-limitative description of embodiments thereof, with reference to theappended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a four-port network composed of coupled-pair TLs;

FIG. 2(a) shows graphs of the even-effective dielectric constant of theeven-mode and the odd-effective dielectric constant of the odd-mode vs.frequency;

FIG. 2(b) shows a graph of the differential group delay of a microstripvs. frequency;

FIG. 3(a) shows graphs of insertion loss vs. frequency for acoupled-pair microstrip;

FIG. 3(b) shows graphs of far-end crosstalk vs. frequency for acoupled-pair microstrip;

FIG. 4(a) shows graphs of insertion loss vs. frequency for severalvalues of distance between microstrips in a coupled-pair microstrip;

FIG. 4(b) shows graphs of far-end crosstalk vs. frequency for severalvalues of distance between microstrips in a coupled-pair microstrip;

FIG. 1(a) and FIG. 5(b) show respectively graphs of Δτ_(MS), andΔτ_(MS), versus frequency;

FIG. 2(a) shows graphs of the approximated crosstalk canceller (XTC) andexact insertion loss (IL) versus frequency for a coupled-pal rmicrostrip;

FIG. 6(b) shows graphs of the approximated crosstalk canceller (XTC) andexact far-end crosstalk (FEXT) versus frequency for a coupled-pairmicrostrip;

FIG. 7 schematically shows an Ultra-broadband optoelectronic systemmodel;

FIG. 8(a), FIG. 8(b), and FIG. 8(c) show graphs of minimum requiredreceived optical power vs. total CP-MS length values for the prior artcrosstalk canceller and the coupling impairments canceller of theinvention;

FIG. 9 schematically shows an extended DSP block for a CP-MS channel;

FIG. 10a shows graphs of power in (P_(in)) vs. fragmentation order Musing the DSP compensation technique on a 25 Gbaud on-off key (OOK)optoelectronic system for different CP-MS lengths;

FIG. 10b , FIG. 10c , and FIG. 10d show graphs of power in (P_(in)) vs.fragmentation order M using different compensation techniques on 25Gbaud OOK, 50 Gbaud OOK, and 25 Gbaud OPAM4 optoelectronic systemsrespectively;

FIG. 11 shows a system block with a coupling impairments pre-compensatorimplemented by a DSP unit at the transmitter side;

FIG. 12 shows a system block with a coupling impairments pre-compensatorimplemented by a DSP unit as in FIG. 11, at the transmitter sideadditionally comprising a DSP unit at the receiver side in order tocompensate for the intersymbol interference (ISI); and

FIG. 13 shows the DSP unit at the receiver side of the system of FIG.12.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

The Invention is methods of compensating for coupling impairments thatoccur in transmission systems comprising two or more transmission lines.The invention will be described herein with reference to a specificexample of optical communication; however it is to be understood thatthe solutions described herein are applicable to any case, e.g.communication applications with rapid transmission lines (above 10giga), in which similar phenomenon occur. In the case of opticalcommunication systems, the next generation of pluggable modules in datacenters connectivity is considered herein. The separation between theoptical components and the electronic blocks introduces a new challengein the transmission of ultra-broadband signals over coupled microstriptraces, which in turn significantly enhances the coupling impairments.In the case of optical systems the goal of the present invention is toprovide inclusive compensation methods for the coupling impairments ofcoupled-pair and multiple pairs of microstrips that will improve thereceiver sensitivity and support significantly longer microstrip tracesas compared to the classical crosstalk compensation technique. Asdiscussed herein below, the compensation methods provided for a pair oftransmission lines can be extended for use with multiple transmissionlines.

Before describing the compensation methods, an exact couplingimpairments model will be derived. This model can be analytically (ornumerically) derived, simulated by simulation tool, and measured In lab.The model is required in order to derive compensation algorithms of theinvention and to measure their performance as compared to existingcompensation methods.

FIG. 1 shows a four-port network composed of coupled-pair transmissionlines (CP-TL) of length L, where each of the ports is terminated withimpedance of Z₀. The studied CP-TL is uniformly coupled, symmetrical,and lossless. In the case that the coupling is absent, each of the TLshas impedance of Z₀ and the frequency-domain voltage waves over the1^(st) and 2^(nd) lines after propagation a distance of Z, respectively,are:V _(out,1) =V _(in,1) e ^(−jβ) ^(TL) ^(z)  (1)andV _(out,2) =V _(in,2) e ^(−jβ) ^(TL) ^(z)  (2)where V_(in,1) and V_(in,2) are the frequency-domain input voltage wavesto the 1^(st) and 2^(nd) TLs, respectively, and β_(TL) is thepropagation constant of a single TL. On the other hand, in the presenceof coupling between the CP-TL, the output voltages in (1)-(2) should bemodified. Following the coupling mode theory, the derivates of (1) and(2) with respect to Z for weak coupling become [7]:

$\begin{matrix}{\frac{d\; V_{{out}\;,1}}{dz} = {{{- j}\;\beta_{TL}V_{{out}\;,1}} - {j\;\kappa_{x}V_{{out},2}}}} & (3) \\{and} & \; \\{{\frac{{dV}_{{out},2}}{dz} = {{{- j}\;\kappa_{x}V_{{out},1}} - {j\;\beta_{TL}V_{{out},2}}}},} & (4)\end{matrix}$where K_(x) is the mutual coupling coefficient between the coupledlines. The solution of the coupled differential system in (3)-(4) forgeneral input wave voltages V_(in,1), and V_(in,2) is [7]:

$\begin{matrix}{\;\begin{matrix}{\mspace{79mu}{V_{{out},1} = {{\frac{V_{{in},1}}{2}( {e^{{- j}\;\beta_{s}z} + e^{{- j}\;\beta_{f}z}} )} + {\frac{V_{{in},2}}{2}( {e^{{- j}\;\beta_{s}z} - e^{{- j}\;\beta_{f}z}} )}}}} \\{= {\lbrack {{V_{{in},1}\mspace{11mu}\cos\mspace{11mu}( {\omega\;\Delta\;\tau_{TL}} )} - {{jV}_{{in},2}\mspace{11mu}\sin\mspace{11mu}( {\omega\Delta\tau}_{TL} )}} \rbrack \cdot e^{{- j}\;\beta_{TL}z}}}\end{matrix}} & (5)\end{matrix}$and similarlyV _(out,2)=[−jV _(in,1) sin(ωΔτ_(TL))+V _(in,2) cos(ωΔτ_(TL))]·e ^(−jβ)^(TL) ^(z)   (6)where the initial conditions at Z=0 are assumed to be the superpositionof the following cases:V _(out,1) =V _(in,1) ,V _(out,2)=0  (7)andV _(out,1)=0,V _(out,2) =V _(in,2)  (8)

In (5)-(6) it was denoted that

$\begin{matrix}{{{\Delta\;\tau_{TL}} = {\frac{\Delta\;\beta_{TL}z}{2\;\omega} = \frac{\kappa_{x}z}{\omega}}},} & (9)\end{matrix}$where Δτ_(TL) is the differential group delay of IL (DGD-TL), w is theangular frequency, Δβ_(TL)=β_(s)−β_(f), β_(s)=β_(TL)+K_(x), andβ_(f)=β_(TL)−K_(x) are the slow and fast propagation constants of theslow and fast propagation modes, respectively. In the case that Δτ_(TL)is non-zero, the slow and the fast propagation modes experience apropagation delay difference that leads to modal dispersion similar topolarization mode dispersion (PMD) in optical fiber. This is furtherelaborated herein below.

To attain the object of the invention the coupling impairments of theCP-TL are studied. The associated coupling transfer function matrix isdefined as follows:

$\begin{matrix}{{{H_{TL}( {j\;\omega} )} = \begin{bmatrix}{{S_{21}( {j\;\omega} )}} & {{- j}{{S_{23}( {j\;\omega} )}}} \\{{- j}{{S_{41}( {j\;\omega} )}}} & {{S_{43}( {j\;\omega} )}}\end{bmatrix}},} & (10)\end{matrix}$where S_(ij)(jω) is the scattering parameter of the i^(th) output andthe j^(th) input of the measured network [8]. From (5)-(6), the couplingtransfer function matrix of CP-TL is:

$\begin{matrix}{{{H_{TL}( {j\;\omega} )} = \begin{bmatrix}{\cos\mspace{11mu}( {\omega\;\Delta\;\tau_{TL}} )} & {{- j}\;\sin\mspace{11mu}( {\omega\;\Delta\;\tau_{TL}} )} \\{{- j}\mspace{11mu}{\sin( {\omega\;\Delta\;\tau_{TL}} )}} & {\cos\mspace{11mu}( {\omega\;\Delta\;\tau_{TL}} )}\end{bmatrix}},} & (11)\end{matrix}$where each of the terms in the main diagonal stands for the insertionloss (IL) of CP-TL and each of the terms in the secondary diagonalstands for the far end crosstalk (FEXT) of CP-TL. The FEXT describes thecoupling between the TLs at the receive end (far end) with respect tothe interfering signal. In the case that ωΔτ_(TL)≠πn, where n isinteger, crosstalk between the two input signals occurs.

The polarization mode dispersion (PMD) impairment is well-known inoptical fiber communication [9]. The random imperfectness and thearbitrary asymmetric structure (e.g., bends, twists, and stresses) alongthe optical fiber randomly change the polarization state of thepropagated light. Thus, each polarization of the light travels at adifferent random speed and pulse broadening is introduced. The 1^(st)order PMD for a single section of optical fiber can be described by theJones matrix for the rotated polarization state [10]:M(θ)=R(−θ)·M·R(θ),  (12)where R(θ) is the linear polarization operator at angle θ which is givenby:

$\begin{matrix}{{R(\theta)} = \begin{bmatrix}{\cos\mspace{11mu}\theta} & {{- \sin}\mspace{11mu}\theta} \\{\sin\mspace{11mu}\theta} & {\cos\mspace{11mu}\theta}\end{bmatrix}} & (13)\end{matrix}$and M is the phase retarder operator which is given by:

$\begin{matrix}{M = {\begin{bmatrix}{\exp( {j\;\omega\;\Delta\;{\tau/2}} )} & 0 \\0 & {\exp( {{- j}\;\omega\;\Delta\;{\tau/2}} )}\end{bmatrix}.}} & (14)\end{matrix}$

In (13), θ is the angle between the state of polarization (SOP) of theelectrical field at the fiber input and the principal state ofpolarization (PSP) of the optical fiber, and in (14) Δτ is thedifferential group delay (DGD). The DGD is random in nature, timevarying and wavelength dependent [9,10]. The worst PMD occurs in thecase of equal projections of the polarization of the electrical field atthe fiber input, i.e. θ=π/4, where the 1^(st) order PMD becomes:

$\begin{matrix}{{M( \frac{\pi}{4} )} = {\begin{bmatrix}{\cos( {\omega\;\Delta\;{\tau/2}} )} & {{- j}\;{\sin( {\omega\;\Delta\;{\tau/2}} )}} \\{{- j}\;{\sin( {\omega\;\Delta\;{\tau/2}} )}} & {\cos( {\omega\;\Delta\;{\tau/2}} )}\end{bmatrix}.}} & (15)\end{matrix}$

This result is mathematically identical to the coupling transferfunction matrix of CP-TL in (11), where Δτ_(TL)=Δτ/2. Therefore, theDGD-TL Δτ_(TL) is fully analogous to the DGD in optical fiber.

Microstrip is commonly used in PCB due to its simple structure and easyintegration compared to other types of PTLs [8]. However, a microstripis an inhomogeneous structure as part of the electromagnetic (EM) fieldpropagates inside the substrate while the rest propagates in thefree-space. In the static limit, all the EM field propagates inside thesubstrate, while the portion of the propagated EM field in thefree-space increases as a function of frequency. Therefore, thesupported mode of propagation In a microstrip is quasi-transverse EM(TEM) [7]. As the ratio between the EM parts is frequency dependent, thedielectric constant of the total propagated EM field, ε_(r) _(eff) (ω),effectively varies with frequency. The effective dielectric constantε_(r) _(eff) (ω) can be calculated by various models [11]. In the casethat the frequency variation of ε_(r) _(eff) (ω) within the signal'sband is significantly pronounced, chromatic dispersion (CD) and signalbroadening are introduced. This results from the non-zero second orderderivative term of the microstrip propagation constant of the injectedsignal. The microstrip propagation constant is given by:

$\begin{matrix}{\mspace{79mu}{{{\beta_{MS}(\omega)} = {\omega\frac{\sqrt{ɛ_{r_{eff}}(\omega)}}{c}}},}} & (16)\end{matrix}$where c is the speed of light. Note that β_(MS)(ω) is equivalent toβ_(TL) and related to the CD [12]. This is a result of the frequencydependence of β_(MS), which is out of the scope of the invention, and itis not included in the coupling model in (11).

In the literature, the CP-MS is commonly analyzed by the normal-modes[7,8]. As the studied CP-MS is symmetrical, the normal modes are theeven-mode, i.e. V_(in,1)=V_(in,2), and the odd-mode, i.e.,V_(in,1)=−V_(in,2). The propagation constants of the even- and odd-modeare denoted as the even propagation constant β_(e) and odd-propagationconstant β_(o). In [7], it has been shown that the coupling model ofsymmetrical CP-TL derived from the coupling mode theory in (11) isidentical to the coupling model derived from the normal-mode method,i.e., β_(e)=β_(s) and β_(o)=β_(f). In quasi-TEM channels such as CP-MS,the mutual-coupling coefficient κ_(x) is not negligible and each of themodes propagates in a different velocity, i.e., Δβ_(TL)≠0 [7].Therefore, in the case of CP-MS, the DGD of microstrip (DGD-MS) Δτ_(MS)is non-zero, which leads to modal dispersion and pulse broadening.

The microstrip propagation constant in (16) can be generalized for theeven-mode propagation constant:

$\begin{matrix}{\mspace{76mu}{{\beta_{e}(\omega)} = {\omega\frac{\sqrt{ɛ_{r_{eff}}^{e}(\omega)}}{c}}}} & (17)\end{matrix}$and for odd-mode propagation constant:

$\begin{matrix}{\mspace{76mu}{{{\beta_{o}(\omega)} = {\omega\frac{\sqrt{ɛ_{r_{eff}}^{o}(\omega)}}{c}}},}} & (18)\end{matrix}$where ε_(r) _(eff) ^(e) is the even-effective dielectric constant of theeven-mode, and ε_(r) _(eff) ^(o) is the odd-effective dielectricconstant of the odd-mode. Therefore, in the case of CP-MS of length L,the DGD-MS is:

$\begin{matrix}{{{\Delta\;\tau_{MS}} = {\frac{\Delta\; n_{MS}}{2\; c}L}},} & (19)\end{matrix}$where Δn_(MS)=√{square root over (ε_(r) _(eff) ^(e) (ω))}−√{square rootover (ε_(r) _(eff) ^(o)(ω))}. Similar to ε_(r) _(eff) , the ε_(r) _(eff)^(e) and ε_(r) _(eff) ^(o) are frequency dependent and can be calculatedby [13]. Additionally, the difference between ε_(r) _(eff) ^(e) andε_(r) _(eff) ^(o) varies with frequency, such that Δn_(MS) is notconstant, and in turn Δτ_(MS) is frequency dependent [13]. In FIG. 2(a),following the theoretical model in [13], and ε_(r) _(eff) ^(o) versusfrequency are presented, and in FIG. 2(b), following (19), Δτ_(MS)versus frequency is shown. Both figures are for CP-MS with L=10 cm,S=100 μm and Z₀=50Ω, which is fabricated over RO3006 substrate(dielectric constant of ε_(r)=6.15) with thickness of 300 μm, which iswithin the range of conventional substrate thickness values [3,14,15].The small analog traces spacing is selected to allow high port density,while lower spacing values of coupled microstrips fabricated over RO3006may be less practical due to fabrication challenges.

FIG. 2(a) and FIG. 2(b) reveal that the frequency dependent effect issignificantly pronounced within the spectrum of the ultra-broadbandelectrical signal. The frequency dependence of Δτ_(MS) within the signalband results in a frequency-dependent modal dispersion. This is similarto the waveguide dispersion in optical fiber, which is in addition tothe PMD [9].

In summary, the coupling impairments of the CP-MS are the following: (a)crosstalk (when the secondary diagonal of the coupling transfer functionis non-zero), (b) modal dispersion (similar to PMD), and (c)frequency-dependent dispersion (similar to waveguide dispersion).

In the case of CP-MS, the coupling transfer function matrix is modifiedto

$\begin{matrix}{{H_{MS}( {j\;\omega} )} = {\begin{bmatrix}{\cos( {\omega\;\Delta\;{\tau_{MS}(\omega)}} )} & {{- j}\;{\sin( {\omega\;\Delta\;{\tau_{MS}(\omega)}} )}} \\{{- j}\;{\sin( {\omega\;\Delta\;{\tau_{MS}(\omega)}} )}} & {\cos( {\omega\;\Delta\;{\tau_{MS}(\omega)}} )}\end{bmatrix}.}} & (20)\end{matrix}$

In FIG. 3(a) and FIG. 3(b) the exact insertion loss (IL) and the exactfar-end crosstalk (FEXT) (as defined in (20)) of CP-MS are presented.The continuous curves are the high frequency structure simulator (HFSS)simulation results, and the dashed curves follow the coupling transferfunction in (20), where the estimated frequency-dependent loss (FDL) is1.6·10⁻² dB/(GHz·cm). Both the continuous and the dashed curves are foridentical parameters of the CP-MS as in FIG. 2(a) and FIG. 2(b). It canbe observed that the simulation results are identical to the proposedcoupling model in (17)-(20). Notice that as Δτ_(MS) is frequencydependent, the IL and the FEXT of the CP-MS follow a quasi-periodicvariation with respect to the frequency. This exact model agrees withthe simulation findings reported in [1]. On the other hand, in the casethat Δτ_(MS) is assumed to be constant, i.e., κ_(x) in (9) isproportionally linear versus frequency, the IL and FEXT obey a periodicvariation, which is presented by the dotted curves of FIGS. 3(a) and 3b, where Δτ_(MS)=Δτ_(MS) (0). The relationship between the exactcoupling model, the approximated periodic coupling model and the XTC isfurther elaborated herein below. Notice that, in the example presentedin FIG. 3(a) and FIG. 3(b), the accuracy of the periodic model islimited to the low frequency region only, i.e., sub-10G transmission.For higher transmission bandwidth (25G and above) a more accurate modelis required. This is the exact coupling model of CP-MS in (17)-(20). Inaddition, in FIG. 4a ) and FIG. 4(b), the exact IL and the exact FEXT(as defined in (20)) of lossless CP-MS versus frequency, for varioustraces spacing S values are presented. All curves are for identicalparameters of the CP-MS as in FIG. 2(a) and FIG. 2(b). FIG. 4(a) andFIG. 4(b) reveal that the quasi-periodic frequency variation effect issignificantly pronounced within the spectrum of the ultra-broadbandelectrical signal.

In recent years, the crosstalk cancellation (XTC) between basebandsignals that are transmitted over coupled microstrips became a standardpractice [3-6]. Those XTC techniques are based on the assumption thatthe signal at the output of the direct line is a linear combination ofthe desired signal with the negative derivative of the coupled signal.Therefore, in the case of lossless CP-MS, the coupling transfer functionmatrix that is assumed for the XTC techniques is [16]:

$\begin{matrix}{{{\hat{H}}_{MS}( {j\;\omega} )} = \begin{bmatrix}1 & {{- j}\;\omega\;\sigma} \\{{- j}\;\omega\;\sigma} & 1\end{bmatrix}} & (21)\end{matrix}$where σ is the forward coupling strength; and recall that the Fouriertransform of the derivative with respect to time is jω. An example ofXTC technique, based on the coupling model in (21), is given in [16].According to this XTC technique, the compensated frequency-domainelectrical signal vector is:

$\begin{matrix}\begin{matrix}{{{\underset{\_}{V}}^{({XTC})}( {j\;\omega} )} = {\begin{bmatrix}1 & {j\;\omega\;\xi} \\{j\;\omega\;\xi} & 1\end{bmatrix}{{\hat{H}}_{MS}( {j\;\omega} )}{V_{in}( {j\;\omega} )}}} \\{= {\begin{bmatrix}{1 + {\sigma\;\xi\;\omega^{2}}} & {{- j}\;{\omega( {\sigma - \xi} )}} \\{{- j}\;{\omega( {\sigma - \xi} )}} & {1 + {\sigma\;\xi\;\omega^{2}}}\end{bmatrix}{V_{in}( {j\;\omega} )}}}\end{matrix} & (22)\end{matrix}$where V_(in) (jω) is the frequency-domain input signal vector, ξ is theXTC gain term, and the underline stands for the compensated signal. Inthe case that ξ=σ, the crosstalk is cancelled and the desired signal isamplified. As a similar coupling model is assumed, all those XTCtechniques are denoted as the “classical-XTC”. Note that the couplingmodel in (21) and in turn the XTC techniques in (22) mainly treat thecrosstalk, while the other coupling impairments, I.e., modal dispersionand frequency-dependent dispersion, are handled narrowly. This limitedtreatment will be discussed in more detail herein below. Therefore, inthe compensation of ultra-broadband electrical signals where all threecoupling impairments are pronounced, the extended coupling model in (20)should be considered.

Herein, the electrical field at the receiver is a linear function of theelectrical field at the transmitter. Thus, the electrical field analysisfollows the classical linear time invariant (LTI) approach. Given thelossless coupling model of CP-MS in (20), the coupling impairments canbe compensated for either in the frequency-domain by the inverse matrixof the sampled coupling model of CP-MS or in the time-domain by theimpulse response of the sampled coupling model of CP-MS. Accordingly, inthe case of frequency-domain compensation, the compensated digitalsignal vector is:

$\begin{matrix}{{v_{- n}^{({EECC})} = {{IDTFT}\{ {{{{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{|t} = \frac{n \cdot T_{s}}{SPS}} \}} \}}},} & (23)\end{matrix}$where n is the sampling index, DTFT stands for the discrete-time Fouriertransform, IDTFT is the inverse DTFT, v_(in) (t,0) is the time-domaininjected ultra-broadband electrical signal column vector at the CP-MSinputs, h_(MS) (t) is the coupling impulse response matrix (i.e.,ℑ⁻¹{H_(MS) (jω)}, where ℑ stands for the Fourier transform), T_(s) isthe sampling period, SPS is the number of samples-per-symbol, * denotesthe convolution operation, Ĥ_(MS) ⁻¹(jω′) is the inverse matrix of thesampled coupling transfer function of CP-MS and given by:

$\begin{matrix}{{{{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} = \begin{bmatrix}{\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )} & {j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} \\{j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} & {\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}\end{bmatrix}},} & (24)\end{matrix}$ω′∈[−ω_(s)/2, ω_(s)/2] is the digital angular frequency, and ω_(s) isthe angular sampling frequency that follows the sampling frequency ofthe analog to digital (ADC), f_(s), which is related to the samplingperiod by T_(s)=2π/ω_(s). Note that H_(MS) (jω′) is a unitary matrix forany ω′Δτ_(MS)(ω′) whose conjugate transpose is also its inverse, i.e.H_(MS) ⁻¹(jω′)=H*_(MS) (jω′). For simplicity of notation, from here onthe prime over ω′ has been dropped. Each of the terms in the maindiagonal stands for the IL of the sampled coupling transfer function ofthe CP-MS and each of the terms in the secondary diagonal stands for theFEXT of the sampled coupling transfer function of the CP-MS. In [17], apreliminary analysis of the basic compensation approach of (23)-(24) ispresented and in [18] a detailed analysis is presented. The compensationtechnique in (23)-(24) is denoted as frequency-domain “exact electricalcoupling compensation” (EECC) for CP-MS.

The EECC in (23)-(24), which is frequency-domain compensation, has beenfurther investigated and an approximated “electrical couplingcompensation” (ECC) technique, which compensates equivalently in thetime-domain and frequency-domain, is derived. In the followingparagraphs, it is shown that using the first order Taylor expansion ofΔτ_(MS)(ω) is suitable for the derivation of the time-domain ECC.Without loss of generality, the Taylor expansion of Δτ_(Ms) (ω) aboutthe central angular frequency ω₀ is:

$\begin{matrix}{{{\Delta\;{\tau_{MS}(\omega)}} = {{\Delta\;\tau_{{MS}_{0}}} + {\Delta\;{\tau_{{MS}_{1}} \cdot ( {\omega - \omega_{0}} )}} + {\frac{\Delta\;\tau_{{MS}_{2}}}{2} \cdot ( {\omega - \omega_{0}} )^{2}} + \ldots}}\mspace{14mu},} & (25)\end{matrix}$where Δτ_(MS) _(m) =(∂^(m)Δτ_(MS)/∂ω^(m))_(|ω=ω) ₀ with m=0, 1, 2, . . .and ∂^(m)/∂ω^(m) is the m^(th) order derivative with respect to thevariable ω. Note that Δτ_(MS) is frequency dependent and high orderterms of the Taylor expansion in (25) are required for the expansion ofΔτ_(MS) in the bandwidth of the ultra-broadband electrical signal. Thiscan be understood from the curves of the first and second derivation ofΔτ_(MS), namely Δτ_(MS), and Δτ_(MS) ₂ .

In FIG. 1(a) and FIG. 5(b), Δτ_(MS) ₁ and Δτ_(MS) ₂ are shown for CP-MSwith L=10 cm S=100 μm, and Z₀=50Ω, which is fabricated over RO3006substrate (ε_(r)=6.15) with thickness of 300 μm. From this example, asΔτ_(MS) ₁ , and Δτ_(MS) ₂ are non-constant within the ultra-broadbandelectrical signal spectrum region, the first and the second order Taylorexpansion about a single frequency is not adequate. Thus, it is proposedto subdivide Δτ_(MS) into multiple sub-bands, where in each sub-bandΔτ_(MS) can be approximated by its associated first order Taylorexpansion. For the k^(th) sub-band of the sampled double-sided signalspectrum of Δτ_(MS), ω∈[−ω_(s)/2, ω_(s)/2], the first order Taylorexpansion about the k^(th) central angular frequency, ω₀ ^((k)), is:

$\begin{matrix}{{\Delta\;{\tau_{MS}^{(k)}(\omega)}} = \{ {\begin{matrix}{{{\Delta\;\tau_{{MS}_{0}}^{(k)}} + {{\Delta\tau}_{{MS}_{1}}^{(k)} \cdot ( {\omega - \omega_{0}^{(k)}} )}},{\omega \in \lbrack {\omega_{\min}^{(k)},\omega_{\max}^{(k)}} \rbrack}} \\{0,{\omega \notin \lbrack {\omega_{\min}^{(k)},\omega_{\max}^{(k)}} \rbrack}}\end{matrix},} } & (26)\end{matrix}$where Δτ_(MS) _(m) ^((k))=(∂^(m) Δτ_(MS)/∂ω^(m))_(|ω=ω) ₀ _((k)) ,ω_(min) ^((k))=ω₀ ^((k))−0.5B^((k)), ω_(max) ^((k))=ω₀^((k))+0.5B^((k)), and B^((k)) is the bandwidth of the k^(th) sub-band.For simplicity, it is assumed that all sub-bands of the double-sidedspectrum have identical bandwidth, i.e., B_(M)=ω_(s)/M, where M is thetotal number of sub-bands of the sampled double-sided spectrum ofΔτ_(MS). Consequently, in the case of an ultra-broadband electricalsignal transmitted over CP-MS, the approximated frequency-domain ECC(which is the approximation of (24)) is given by:

$\begin{matrix}{{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} = \begin{bmatrix}{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} & {- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} \\{- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} & {{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )}\end{bmatrix}},} & (27)\end{matrix}$where the approximated IL term is

$\begin{matrix}{{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} = {\cos( {\omega{\sum\limits_{k = 1}^{M}\;{\Delta\;{\tau_{MS}^{(k)}(\omega)}}}} )}} & (28)\end{matrix}$and the approximated FEXT term is

$\begin{matrix}{{{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} = {{- j}\;{{\sin( {\omega{\sum\limits_{k = 1}^{M}\;{\Delta\;{\tau_{MS}^{(k)}(\omega)}}}} )}.}}} & (29)\end{matrix}$

This compensation method is denoted as the frequency-domain spectrallyfragmented-ECC (SF-ECC) for CP-MS. Consequently, the compensated digitalsignal vector is:

$\begin{matrix}{v_{- n}^{({{SF} - {ECC}})} = {{IDTFT}{\{ {{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{|t} = \frac{n \cdot T_{s}}{SPS}} \}} \}.}}} & (30)\end{matrix}$

Note that in the case of M→∞, the SF-ECC converges to the EECC for CP-MSin (23)-(24). In addition, the coupling impairments can be compensatedfor in the time-domain by the impulse response of the sampled couplingmodel of CP-MS. Given the SF-ECC in (27)-(30), the time-domain SF-ECCis:

$\begin{matrix}{{{{\overset{\sim}{h}}_{MS}^{- 1}(t)} = \begin{bmatrix}{{\overset{\sim}{h}}_{IL}^{({MS})}(t)} & {- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} \\{- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} & {{\overset{\sim}{h}}_{IL}^{({MS})}(t)}\end{bmatrix}},} & (31)\end{matrix}$where the impulse response of the approximated IL term is:

h ~ IL ( MS ) ⁢ ( t ) = ⁢ - 1 ⁢ { H ~ IL ( MS ) ⁡ ( j ⁢ ⁢ ω ) } = ⁢ ∑ k = 1 M ⁢ ⁢h ~ IL ( MS ) ⁢ ( k ) ⁡ ( t ) ( 32 )and the impulse response of the approximated FEXT term is:

h ~ FEXT ( MS ) ⁢ ( t ) = ⁢ - 1 ⁢ { H ~ FEXT ( MS ) ⁡ ( j ⁢ ⁢ ω ) } = ⁢ ∑ k = 1M ⁢ ⁢ h ~ FEXT ( MS ) ⁢ ( k ) ⁡ ( t ) . ( 33 )

The detailed derivation of (32)-(33) is given in an Appendix locatedbefore the Bibliography, and {tilde over (h)}_(IL) ^((MS)(k))(t) and{tilde over (h)}_(FEXT) ^((MS)(k))(t) are the impulse responses of theapproximated IL and FEXT terms within the k^(th) sub-band, respectively.The impulse response of the IL of CP-MS within the k^(th) sub-band is:

h ~ IL ( MS ) ⁢ ( k ) ⁡ ( t ) = ⁢ - 1 ⁢ { H ~ IL ( MS ) ⁢ ( k ) ⁡ ( j ⁢ ⁢ ω ) }= ⁢ 1 2 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ cos ⁡ ( ω ⁢ ⁢ Δ ⁢ ⁢ τ MS ⁡ ( ω ) ) ⁢ Π ⁡ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0( k ) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω ≅ ⁢ 1 2 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ cos ⁡ ( ω ⁢ ⁢ Δ ⁢ ⁢ τ MS (I ) ( k ) ⁡ ( ω ) ) ⁢ Π ⁡ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k ) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢1 4 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ e j ⁢ ⁢ ω ⁡ ( Δ ⁢ ⁢ τ MS 0 ( k ) - ω 0 ( k ) ⁢ Δ ⁢ ⁢ τ MS 1 (k ) ) + j ⁢ ⁢ ω 2 ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ⁢ Π ⁢ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k ) ) ⁢ e j ⁢ ⁢ ω⁢⁢t ⁢ ⁢ d ⁢ ⁢ ω + ⁢ 1 4 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ e - j ⁢ ⁢ ω ⁡ ( Δ ⁢ ⁢ τ MS 0 ( k ) - ω 0 ( k )⁢Δ ⁢ ⁢ τ MS 1 ( k ) ) - j ⁢ ⁢ ω 2 ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ⁢ Π ⁢ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢ A · [ j ⁢ e - j ⁢ ⁢ ϑ ( k ) ⁡ ( t ) * δ ⁡ ( t + γ( k ) ) + - j ⁢ e j ⁢ ⁢ ϑ ( k ) ⁡ ( t ) * δ ⁡ ( t - γ ( k ) ) ] * sin ⁢ ⁢ c ⁡ (B M 2 ⁢ t ) ⁢ e j ⁢ ⁢ ω 0 ( k ) ⁢ t , ( 34 )where {tilde over (H)}_(IL) ^((MS)(k))(jω) is the IL of CP-MS within thek^(th) sub-band, Δτ_(MS) ₍₁₎ ^((k)) (ω)=Δτ_(MS) ₀ ^((k))+Δτ_(MS) ₁^((k)) (ω−ω₀ ^((k))) is the first order Taylor expansion of the k^(th)sub-band, A=B_(M)/(8π√{square root over (πΔτ_(MS) ₁ ^((k)))})γ^((k))=Δτ_(MS) ₀ ^((k))−ω₀ ^((k))Δτ_(MS) ₁ ^((k)), and ϑ^((k))(t)=t²/(4Δτ_(MS) ₁ ^((k))). Also, the following inverse Fouriertransforms have been used in (34):

⁢ - 1 ⁢ { e ± j ⁢ ⁢ ω ⁢ ⁢ γ ( k ) } = δ ⁡ ( t ± γ ( k ) ) ⁢ ⁢ and ( 35 ) - 1 ⁢ { e± j ⁢ ⁢ ω 2 ⁢ Δ ⁢ ⁢ τ MS 1 ( t ) } = ± j 4 ⁢ ⁢ π ⁢ ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ⁢ e ∓ jt 2 /( 4 ⁢ ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ) = ± j 4 ⁢ ⁢ π ⁢ ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ⁢ e ∓ j ⁢ ⁢ ϑ ( k )⁡( t ) , ( 36 )

Similarly, the k^(th) impulse response of the approximated FEXT is:

$\begin{matrix}{{{\overset{\sim}{h}}_{FEXT}^{{({MS})}{(k)}}(t)} \cong {{A \cdot \begin{bmatrix}{{{- \sqrt{j}}e^{{- j}\;{\vartheta^{(k)}{(t)}}}*{\delta( {t + \gamma^{(k)}} )}} +} \\{\sqrt{- j}e^{{- j}\;{\vartheta^{(k)}{(t)}}}*{\delta( {t - \gamma^{(k)}} )}}\end{bmatrix}}*\sin\;{c( {\frac{B_{M}}{2}t} )}e^{j\;\omega_{0}^{(k)}t}}} & (37)\end{matrix}$and the compensated digital signal vector is:

${\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {\{ {{{\overset{\sim}{h}}_{MS}^{- 1}(t)}*\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{t = \frac{n \cdot T_{s}}{SPS}}} \}} \}.}$

Note that the approximation in the third line of (34) is justified dueto the assumption that the first order Taylor expansion about ω₀ ^((k))is sufficient in B_(M), I.e., the fragmentation order M is properlyselected. The substitution of (34) in (32) and (37) in (33) forms thetime-domain SF-EDC equalizer, which includes a filter-bank of M parallelequalizers. Recall that each of the M parallel equalizers iscompensating for the k^(th) sub-bands of the ultra-broadband signal.Hence, the compensation can be performed equivalently in thefrequency-domain using (27)-(30) or in the time-domain using (31)-(37).

Assuming that Δτ_(MS) (ω) is constant within the transmission band suchthat Δτ_(MS) (ω)≅Δτ_(MS) (0), then the IL term in (20) is reduced to:H _(IL) ^((MS))(jω)=cos(ωΔτ_(MS)(0))  (38)and the FEXT term in (20) is reduced to:H _(FEXT) ^((MS))(jω)=−j sin(ωΔτ_(MS)(0)).  (39)In turn the IL and FEXT terms in (38)-(39) follow periodic variationwith respect to frequency. Note that the frequency dependence of Δτ_(MS)is affected by the transmission band, CP-MS length, L, and its couplingstrength, κ_(x). Therefore, this assumption usually holds for sub-10Gtransmission, as can be observed from the periodic curves (constantΔτ_(MS)) compared with the exact curves (frequency dependent Δτ_(MS)) inFIG. 3(a) and FIG. 3(b), or when the DSP block is part of the hybridoptical-electrical module, i.e., a short microstrip length of a few cm.

In case where a small-angle approximation of the IL in (38) and the FEXTin (39) can be applied, the coupling transfer function matrix reducesto:

$\begin{matrix}{{{{\overset{\Cup}{H}}_{MS}( {j\;\omega} )} = \begin{bmatrix}1 & {{- j}\;\omega\;\Delta\;{\tau_{MS}(0)}} \\{{- j}\;\omega\;\Delta\;{\tau_{MS}(0)}} & 1\end{bmatrix}},} & (40)\end{matrix}$

This result is identical to the coupling transfer function matrix in(21) with σ=Δτ_(MS) (0), which is the basis for the “classical-XTC”technique. In FIG. 6(a) and FIG. 6(b), the IL and FEXT of the XTC'scoupling model (as defined in (21)), which is denoted as approximated ILand approximated FEXT, respectively, and the exact IL and exact FEXT(using (20)) are presented. The curves are for CP-MS with L=5 cm, S=100μm, Z₀=50Ω, ε_(r)=6.15 and substrate thickness of 300 μm. In the case ofthe XTC's coupling model, it was assumed that σ=Δτ_(MS) (0). Thiscomparison reveals that the XTC's coupling model and in-turn the XTConly holds for limited transmission bandwidth, limited CP-MS length, andlimited CP-MS coupling strength. For ultra-broadband communication links(25G and above), the ECC techniques of the invention, i.e. EECC,time-domain SF-ECC, frequency-domain SF-ECC, and 3ECC, which are basedon the exact coupling model, should be used.

In FIG. 7, the block diagram of a pair of 25-50 Gbaud non-return-to-zero(NRZ) OOK (50-100 Gb/sec) or a pair of 25 Gbaud optical PAM4 (100Gb/sec) transmission systems are presented. Depending on the signalmodulation, the digital data at the client side is converted into analogsignal by a 1-bit or 2-bit digital to analog converter (DAC) for OOK orPAM4, respectively, that has a Butterworth filter response withfrequency cutoff (and order) of 17 GHz (3^(rd)). The electrical signalis transmitted over a CP-MS; where its model is given in (17)-(20). TheCP-MS parameters are Z₀=50Ω, S=100 μm, ε_(r)=6.15, substrate thicknessof 300 μm, and variable traces length L. At the CP-MS output, each ofthe client side signals is converted to an optical signal using anelectrical to optical converter (EOC) that consists of a modulatordriver and a modulator with extinction ratio of 6 dB. The driver and themodulator have a Butterworth filter response with frequency cutoff (andorder) of 25 GHz (3^(rd)) and 30 GHz (2^(nd)), respectively. In turn,the optical signals are combined by an optical multiplexer (MUX) and aretransmitted over a very short optical fiber (sub 2 km), such that it maybe treated as an optical back-to-back transmission. At the receiverside, the optical signal is demultiplexed (via an optical DEMUX)followed by a variable optical attenuator (VOA) and each of the opticalchannels is detected by an optical to electrical converter (OEC). TheOEC consists of a photodiode with responsivity of 0.7 [A/W] andtrans-impedance amplifier (TIA) including automatic gain control (AGC).The noise source following the photodiode is assumed to obey an additivewhite Gaussian noise (AWGN) model with zero mean and 14 [pA/√{squareroot over (Hz)}]. The electrical noised signal is filtered by the OEC'sblocks. Each of the OEC's blocks, i.e., photodiode and TIA, has aButterworth filter response with frequency cutoff (and order) of 30 GHz(2^(nd)), and 25 GHz (3^(rd)). The filtered data is transmitted overCP-MS with parameters identical to those of the CP-MS in the transmitterside. At the CP-MS output, each of the two electrical signals is sampledby an analog to digital converter (ADC) with sampling rate of 50 Gsamples/sec and an effective number of bits (ENOB) of 5 bits. The ADChas a Butterworth filter response with cutoff frequency (and order) of17 GHz (3^(rd)). Finally, the sampled signal is post-processed by theDSP block, which compensates for the entire CP-MS traces (both In thetransmitter- and receiver-side), and decoded. As this is intensitymodulation and direct-detection (IM\DD) scheme, it preserved the linearrelationship between the electrical modulated signal at the transmitterinput and the electrical signal at the receiver output.

An inclusive set of Monte-Carlo simulations was performed, and thedifferent compensation techniques were analyzed and compared. In thesimulation, the entire channel modelling including the optoelectroniccomponents, and the electrical traces is based on the analytical modelsthat are presented above. The optical channel is modelled asback-to-back transmission. The parameters being used in theoptoelectronic models are based on off-the-shelf components, e.g., ENOBand bandwidth. The analysis includes the following compensatingtechniques: (a) EECC (exact electrical coupling compensation), which isdefined In equations (27)-(30), (b) 3ECC (enhanced exact electricalcoupling compensation), which is defined further herein below, and (c)SF-ECC (spectrally fragmented electrical coupling compensation), whichis defined in equations (31)-(37). All these compensation techniques arecompared to the classical-XTC and an enhanced version of it (EXTC),which is defined further herein below. All the compensation filters areimplemented in the frequency domain following a 50G samples/sec ADC. Thedata stream includes two cases of 25 Gbaud and 50 Gbaud, with2-samples-per-symbol (SPS) and 1-SPS, respectively. This 2-SPS and 1-SPSfrequency-domain equalization is equivalent to T/2 and T spacedtime-domain FIR equalizer, respectively.

The performance of the EECC method of the invention versus“classical-XTC” is analyzed by means of the minimum required receivedoptical power, which is denoted as P_(in), to achieve pre-forward errorcorrection (pre-FEC) bit error rate (BER) value of 10⁻³. This BER valueis achieved by selecting the appropriate optical power, which is tunedby the VOA. The P_(in) is measured for various total CP-MS length valuesL_(total)=2L (including both portions at the transmitter and receiver)and the results are summarized in FIG. 8(a)-8(c). FIG. 8(a) depicts theresults for the 25 Gbaud OOK, where each of the curves represents adifferent compensation technique. In addition, the uncompensated data Isshown by the “dotted-curve”. The compensation of the couplingimpairments by the “classical-XTC” and EECC are shown by the“continuous-curve” and “squares-curve”, respectively. It can be observedthat the “classical-XTC” slightly improves the system performance ascompared to the uncompensated data. Yet, the XTC is limited to veryshort traces (heading a brick wall at 3 cm). On the other hand, the EECCmethod of the invention improves the system performance and supportssignificantly longer CP-MS.

In contrast to multiple input multiple output (MIMO) channels withrandom coupling impairments, such as PMD in optical fiber, that areinherently limited by the channel approximation, and are compensated bystochastic equalization, e.g., maximum likelihood sequence estimation(MLSE), here the coupling compensation techniques of the invention arepure fixed and deterministic compensation for the deterministic couplingimpairments of the CP-MS, which is similar to the compensation ofchromatic dispersion in optical fiber. However, the intersymbolinterference (ISI) that is introduced by the limited bandwidth of theDAC and the ADC is not compensated. Therefore, in order to mitigate thecombined coupling impairments of the CP-MS and ISI distortionsassociated with the bandlimited components, the DSP engine at thereceiver side Is extended and comprises an additional interpolatorfollowed by a feed-forward equalizer and decision feedback equalizer(FFE-DFE). The extended DSP block is shown in FIG. 9, where the couplingimpairments compensator can be realized either by XTC or by EECC.Depending on the coupling Impairments compensator, the extended DSP isdenoted as enhanced XTC (EXTC) or enhanced EECC (3ECC). The interpolatorretrieves the optimal time-domain sampling phase and the FFE-DFEadaptively mitigates the ISI penalty by using the least mean square(LMS) algorithm. In FIG. 8(a), the compensation of coupling Impairmentsby XTC or ECC, are compared with the EXTC and 3ECC. The performanceanalysis of 25 Gbaud OOK by using EXTC and 3ECC is presented by the“circles-curve” and “diamonds-curve”, respectively. For all CP-MSlengths, the performances of EXTC and 3ECC are significantly Improved ascompare to the XTC and EECC techniques. The results indicate that theEXTC technique supports longer L_(total) values than the XTC, yet itsupports significantly shorter traces lengths as compared to the 3ECCtechnique.

Additionally, the extended DSP engine is used for analyzing two 50Gbit/sec systems, while the systems' hardware elements remain the same:(1) a 50 Gbaud OOK system, where the transmitted data rate is doubled,and (2) a 25 Gbaud optical 4-ary pulse amplitude modulation (PAM4). Theresults are summarized in FIG. 8(b) and FIG. 8(c), respectively. Thecurves Indicate that the proposed 3ECC improves the system performanceand supports significantly longer CP-MS, while the EXTC is headingtowards a “brick wall” after relatively short length of a few cm.

Furthermore, the SF-ECC technique is analyzed. The sampled signal iscompensated by the DSP block that consists of the frequency-domainSF-ECC, while the other system components remain the same. The impact ofthe fragmentation order M on the 25 Gbaud OOK is studied and the resultsare summarized in FIG. 10(a), where each curve represents a differenttotal CP-MS length L_(total), and the back-to-back scenario is denotedas L_(total)=0. As the number of fragmented bands M Increases, therequired P_(in) decreases, indicating the improvement introduced by thefrequency-domain SF-ECC. The optimal value of M, which is associatedwith the knee point of the P_(in) curves, indicates the number ofrequired sub-bands of the frequency-domain equalizer in (27) orequivalently the number of parallel filter-bank equalizers in (31). Thebest improvement is achieved with M=5 and essentially for M>5 theperformance of the SF-ECC is identical to the performance of the EECCthat is shown in FIG. 8(a) by the “squares-curve”.

Finally, the new generation of wide band optics (25-30G), e.g., siliconphotonics based components, that leads to overall channel bandwidth of25 GHz and above is considered. Therefore, the frequency responses ofthe DAC, EOC, OEC and ADC are modelled by an equivalent frequencyresponse, which obeys a Butterworth filter with an overall frequencycutoff of 25 GHz and order of 6 and 3 for the signal and noise,respectively. The other system parameters remain the same and the DSPblock consists of EXTC or 3ECC. The analysis results of 25 Gbaud OOK, 50Gbaud OOK, and 25 Gbaud OPAM4 are presented in FIG. 10(b)-(d),respectively. The “circles-curves” and “diamonds-curves” stand for theEXTC and 3ECC techniques, respectively. For all CP-MS lengths, theP_(in) performances of EXTC and 3ECC are improved as compare to theperformance in FIG. 8. Yet, the 3ECC technique supports significantlylonger L_(total) values as compared to the EXTC.

To summarize all of the above, an exact coupling model of CP-MS wasdeveloped and revealed that for ultra-broadband communication links (25Gand above) the XTC's coupling model is not valid. Therefore, theclassical-XTC technique is limited to sub-10G transmission or shortmicrostrip length. On the other hand, the compensation techniques of thecurrent invention, i.e., EECC, 3ECC and SF-ECC, which are based on theexact coupling model, support longer traces length. In the case of 100GPAM4 transmission and by using commercially available 25G components,the new 3ECC technique supports microstrip traces of 15 cm, while EXTCIs limited to 5 cm. Additionally, the new generation of siliconphotonics components is considered herein and it is shown that even inthis case the new 3ECC technique supports significantly longermicrostrip traces compared to EXTC.

The EECC algorithm, which is described by equations (23) can be utilizedfor coupled multiple transmission lines (more than two lines) by usingthe exact Inverse matrix sampled coupling transfer function of thecoupled multiple transmission lines. Similarly, the SF-ECC algorithm,which is described by (30), can be utilized for coupled multipletransmission lines by using the approximated frequency-domain transferfunction of the coupled multiple transmission lines, which is theapproximation of the exact inverse matrix sampled coupling transferfunction of the coupled multiple transmission lines, or by using theapproximated time-domain impulse response of coupled multipletransmission lines. The exact coupling transfer function matrix ofspecific coupled multiple transmission lines can be calculated,simulated or measured, and then be used for the derivation of the exactinverse matrix sampled coupling transfer function, the approximatedfrequency-domain transfer function, and the approximated time-domainimpulse response of the coupled multiple transmission lines. Thisderivation is similar to the derivation of the transfer function andimpulse response matrix that have been demonstrated herein above for theEECC and SF-ECC of CP-MS. Additionally, the DSP engine at the receiverside can be extended and consists of additional interpolator followed byFFE-DFE, in order to compensate for the ISI. The extended block is shownin FIG. 9, where the coupling impairments compensation can be realizedby either the EECC or the SF-ECC of coupled (pair or multiple)transmission lines.

The invention is Implemented by different embodiments of DSP chips thatare designed to realize one or more of the algorithms of the invention.The exact design of the DSP chip is dependent on the application. Hencethe DSP chip Is sometimes defined as an application-specific integratedcircuit (ASIC) DSP chip. Each of the coupling impairments compensationalgorithms in this invention is realized by a DSP chip. The DSP chip canbe different for each algorithm or generic for more than one of thealgorithms with the option of selecting between them, or part of a DSPchip that performs additional digital signal processing tasks. Hence,the DSP chips comprise a coupling impairments compensator that can berealized by each of the coupling impairment compensation algorithms ofthis invention i.e., EECC (equations (23)-(24)), time-domain SF-ECC(equations (31)-(37)), frequency-domain SF-ECC (equations (27)-(30)) forpair (or multiple) coupled microstrips. The coupling impairmentscompensator can be used for either pre-compensation or forpost-compensation of the coupling impairments as will be discussedherein below. Additionally, the OSP block can be extended as describedherein above in order to compensate for the intersymbol interference(ISI) in which case, the block diagram of FIG. 9 is used forpost-compensation (3ECC).

Algorithms of the invention can be implemented either forpre-compensation, i.e. on the transmitter side of the transmissionsystem, as shown in FIG. 11, or for post-compensation, i.e. on thereceiver side, as shown in FIG. 7, or on both transmitter and receiversides as shown In FIG. 12.

FIG. 11 shows a system block with a coupling impairments pre-compensatorimplemented by a DSP unit at the transmitter side. The DSP chipcomprises a coupling impairments compensator that is realized by eitherthe EECC or the SF-ECC compensation algorithm of the invention forcoupled (pair or multiple) transmission lines.

FIG. 12 shows a system block with a coupling Impairments pre-compensatorimplemented by a DSP unit as in FIG. 11, at the transmitter sideadditionally comprising a DSP engine, which comprises a couplingimpairments compensator that is realized by either the EECC or theSF-ECC compensation algorithm of the invention for coupled (pair ormultiple) transmission lines and an interpolator followed by FFE-DFE (asshown in FIG. 13), which is added at the receiver side in order tocompensate for the ISI.

Although embodiments of the invention have been described by way ofillustration, it will be understood that the invention may be carriedout with many variations, modifications, and adaptations, withoutexceeding the scope of the claims.

APPENDIX

The derivation of (32) is:

h ~ IL ( MS ) ⁢ ( k ) ⁡ ( t ) = ⁢ - 1 ⁢ { H ~ IL ( MS ) ⁡ ( j ⁢ ⁢ ω ) } = ⁢ 1 2 ⁢⁢π ⁢ ∫ - ∞ ∞ ⁢ H ~ IL ( MS ) ⁡ ( j ⁢ ⁢ ω ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢ 1 2 ⁢ ⁢ π ⁢ ∑k = 1 M ⁢ ⁢ ∫ ω 0 ( k ) - B M 2 ω 0 ( k ) + B M 2 ⁢ H ~ IL ( MS ) ⁡ ( j ⁢ ⁢ ω) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢ 1 2 ⁢ ⁢ π ⁢ ∑ k = 1 M ⁢ ⁢ ∫ - ∞ ∞ ⁢ H ~ IL ( MS ) ⁡ (j ⁢ ⁢ ω ) ⁢ Π ⁡ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k ) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢ ∑ k = 1 M ⁢⁢ - 1 ⁢ { H ~ IL ( MS ) ⁢ ( k ) ⁡ ( j ⁢ ⁢ ω ) } = ⁢ ∑ k = 1 M ⁢ ⁢ h ~ IL ( MS ) ⁢( k ) ⁡ ( t ) ( 41 )where {tilde over (H)}_(IL) ^((MS)(k))(jω) is the IL of CP-MS within thek^(th) IL sub-band, {tilde over (h)}_(IL) ^((MS)(k))(t) is the impulseresponse of the IL of CP-MS within the k^(th) sub-band, Π is therectangular function, which defined as:

$\begin{matrix}{{\Pi( {j\;\omega} )} = \{ \begin{matrix}{0,{{\omega } > \frac{B_{M}}{2}}} \\{1,{{\omega } \leq \frac{B_{M}}{2}}}\end{matrix} } & (42)\end{matrix}$and the impulse response of Π is;

 - 1 ⁢ { Π ⁡ ( j ⁢ ⁢ ω ) } = 1 π ⁢ ⁢ t ⁢ sin ⁡ ( B M 2 ⁢ t ) = B M 2 ⁢ ⁢ π ⁢ sin ⁢ ⁢ c⁡( B M 2 ⁢ t ) . ( 43 )

Note that a similar approach is taken in the calculation of the impulseresponse of the {tilde over (H)}_(FEXT) ^((MS))(jω).

BIBLIOGRAPHY

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The invention claimed is:
 1. A digital signal processing (DSP) unit foran ultra-broadband optoelectronic system comprising two or more coupledpair-microstrip (CP-MS) traces, the DSP unit comprising an electroniccircuit configured to realize at least one of an EECC algorithm, afrequency-domain SF-ECC algorithm, and a time-domain SF-ECC algorithm;the DSP characterized in that: a) the EECC algorithm is:${\underset{\_}{v}}_{n}^{({EECC})} = {{IDTFT}\{ {{{{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{t = \frac{n \cdot T_{s}}{SPS}}} \}} \}}$wherein v _(n) ^((EECC)) is the compensated digital signal vector, n isthe sampling index, DTFT stands for the discrete-time Fourier transform,IDTFT is the inverse DTFT, V_(in)(t,0) is the time-domain injectedultra-broadband electrical signal column vector at the CP-MS inputs,h_(ms)(t) is the coupling impulse response matrix ℑ⁻¹{H_(MS) (jω)}, ℑstands for the Fourier transform, T_(s) is the sampling period, SPS isthe number of samples-per-symbol, * denotes the convolution operation,Ĥ_(MS) ⁻¹(jω′) is the inverse matrix of the sampled coupling transferfunction, ω′∈[−ω_(s)/2, ω_(s)/2] is the digital angular frequency, andω_(s) is the angular sampling frequency that follows the samplingfrequency f_(s) of the analog to digital (ADC), which is related to thesampling period by T_(s)=2π/ω_(s); and b) the frequency domain SF-ECCalgorithm is:${{\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {{IDTFT}\{ {{\overset{\sim}{H}}_{MS}^{- 1}{( {j\;\omega} ) \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{t = \frac{n \cdot T_{s}}{SPS}}} \}} \}}};$wherein the DSP unit is extended to compensate for intersymbolinterference (ISI) by addition of an interpolator and a feed-forwardequalizer and decision-feedback equalizer (FFE-DFE).
 2. The DSP of claim1, wherein the inverse matrix of the sampled coupling transfer functionis:${{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} = {\begin{bmatrix}{\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )} & {j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} \\{j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} & {\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}\end{bmatrix}.}$
 3. The DSP of claim 1, wherein the frequency-domainSF-ECC algorithm comprises:${{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} = {\begin{bmatrix}{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} & {- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} \\{- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} & {{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )}\end{bmatrix}\mspace{14mu}{where}}},{{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} = {{\cos( {\omega{\sum\limits_{k = 1}^{M}{\Delta\;{\tau_{MS}^{(k)}(\omega)}}}} )}\mspace{14mu}{and}}}$${{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} = {{- j}\;{\sin( {\omega{\sum\limits_{k = 1}^{M}{\Delta\;{\tau_{MS}^{(k)}(\omega)}}}} )}}$and the compensated digital signal vector is:${\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {{IDTFT}{\{ {{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{❘t} = \frac{n \cdot T_{s}}{SPS}} \}} \}.}}$4. The DSP of claim 1, wherein the time-domain SF-ECC algorithmcomprises:$\mspace{20mu}{{{\overset{\sim}{h}}_{MS}^{- 1}(t)} = \begin{bmatrix}{{\overset{\sim}{h}}_{IL}^{({MS})}(t)} & {- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} \\{- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} & {{\overset{\sim}{h}}_{IL}^{({MS})}(t)}\end{bmatrix}}$   where $\mspace{79mu}\begin{matrix}{{{{\overset{\sim}{h}}_{IL}^{({MS})}(t)} = {{\mathfrak{J}}^{- 1}\{ {{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} \}}},} \\{= {\sum\limits_{k = 1}^{M}{{\overset{\sim}{h}}_{IL}^{{({MS})}{(k)}}(t)}}}\end{matrix}$ $\mspace{20mu}\begin{matrix}{{{{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)} = {{\mathfrak{J}}^{- 1}\{ {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} \}}},} \\{= {\sum\limits_{k = 1}^{M}{{\overset{\sim}{h}}_{FEXT}^{{({MS})}{(k)}}(t)}}}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{h}}_{IL}^{{({MS})}{(k)}}(t)} = {{\mathfrak{J}}^{- 1}\{ {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} \}}} \\{= {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{\cos( {{\omega\Delta\tau}_{MS}(\omega)} )}\ {\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}\; d\;\omega}}}} \\{\cong {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{\cos( {{\omega\Delta\tau}_{{MS}{(I)}}^{(k)}(\omega)} )}\ {\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}\; d\;\omega}}}} \\{= {{\frac{1}{4\pi}{\int_{- \infty}^{\infty}{e^{{j\;{\omega{({{\Delta\tau}_{{MS}_{0}}^{(k)} - {\omega_{0}^{(k)}{\Delta\tau}_{{MS}_{1}}^{(k)}}})}}} + {j\;\omega^{2}{\Delta\tau}_{{MS}_{1}}^{(k)}}}{\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}d\;\omega}}} +}} \\{\frac{1}{4\pi}{\int_{- \infty}^{\infty}{e^{{j\;{\omega{({{\Delta\tau}_{{MS}_{0}}^{(k)} - {\omega_{0}^{(k)}{\Delta\tau}_{{MS}_{1}}^{(k)}}})}}} + {j\;\omega^{2}{\Delta\tau}_{{MS}_{1}}^{(k)}}}{\Pi( {{j\;\omega} - {j\;\omega_{0}^{(k)}}} )}e^{j\;\omega\; t}d\;\omega}}} \\{= {{A \cdot \begin{bmatrix}{{\sqrt{j}e^{{- j}\;{\vartheta^{(k)}{(t)}}}*{\delta( {t + \gamma^{(k)}} )}} +} \\{\sqrt{- j}e^{j\;{\vartheta^{(k)}{(t)}}}*{\delta( {t + \gamma^{(k)}} )}}\end{bmatrix}}*\sin\;{c( {\frac{B_{M}}{2}t} )}e^{j\;\omega_{0}^{(k)}t}}}\end{matrix}$   and$\mspace{79mu}{{{\overset{\sim}{h}}_{FEXT}^{{({MS})}{(k)}}(t)} \cong {{A \cdot \begin{bmatrix}{{\sqrt{j}e^{{- j}\;{\vartheta^{(k)}{(t)}}}*{\delta( {t + \gamma^{(k)}} )}} +} \\{\sqrt{- j}e^{j\;{\vartheta^{(k)}{(t)}}}*{\delta( {t + \gamma^{(k)}} )}}\end{bmatrix}}*\sin\;{c( {\frac{B_{M}}{2}t} )}e^{j\;\omega_{0}^{(k)}t}}}$and the compensated digital signal vector is:${\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {\{ {{{\overset{\sim}{h}}_{MS}^{- 1}(t)}*\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{❘t} = \frac{n \cdot T_{s}}{SPS}} \}} \}.}$5. The DSP of claim 1, wherein the EECC algorithm can be utilized forcoupled multiple transmission lines by using the exact inverse matrixsampled coupling transfer function of the coupled multiple transmissionlines.
 6. The DSP of claim 1, wherein the SF-ECC algorithm can beutilized for coupled multiple transmission lines by using theapproximated frequency-domain transfer function of the coupled multipletransmission lines or by using the approximated time-domain impulseresponse of coupled multiple transmission lines.
 7. An ultra-broadbandoptoelectronic system comprising two or more coupled pair-microstrip(CP-MS) traces and a digital signal processing (DSP) unit comprising anelectronic circuit configured to compensate for coupling impairments byrealizing at least one of an exact electrical coupling impairmentscompensation (EECC) algorithm and a spectrally fragmented electricalcoupling compensation (SF-ECC) algorithm, wherein the SF-ECC can be usedfor compensation of coupling impairments in the frequency domain by afrequency-domain SF-ECC algorithm, and in the time domain by atime-domain SF-ECC algorithm; the system characterized in that: a) theEECC algorithm comprises:${\underset{\_}{v}}_{n}^{({EECC})} = {{IDTFT}\{ {{{{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{❘t} = \frac{n \cdot T_{s}}{SPS}} \}} \}}$wherein v _(n) ^((EECC)) is the compensated digital signal vector, n isthe sampling index, DTFT stands for the discrete-time Fourier transform,IDTFT is the inverse DTFT, V_(in)(t,0) is the time-domain injectedultra-broadband electrical signal column vector at the CP-MS inputs,h_(ms)(t) is the coupling impulse response matrix ℑ⁻¹ {H_(MS) (jω)}, ℑstands for the Fourier transform, T_(s) is the sampling period, SPS isthe number of samples-per-symbol, * denotes the convolution operation,Ĥ_(MS) ⁻¹(jω′) is the inverse matrix of the sampled coupling transferfunction, ω′∈[−ω_(s)/2, ω_(s)/2] is the digital angular frequency, andω_(s) is the angular sampling frequency that follows the samplingfrequency f_(s) of the analog to digital (ADC), which is related to thesampling period by T_(s)=2π/ω_(s); and b) the frequency domain SF-ECCalgorithm is:${{\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {{IDTFT}\{ {{{{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{❘t} = \frac{n \cdot T_{s}}{SPS}} \}} \}}};$wherein the DSP unit is located at one of: a) the transmitter side ofthe optoelectronic system for pre-compensation implementation of thecoupling impairments by the EECC or SF-ECC algorithms; and b) thereceiver side of the optoelectronic system for post-compensationimplementation of the coupling impairments by the EECC or SF-ECCalgorithms.
 8. The system of claim 7, wherein the inverse matrix of thesampled coupling transfer function is:${{\hat{H}}_{MS}^{- 1}( {j\;\omega^{\prime}} )} = {\begin{bmatrix}{\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )} & {j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} \\{j\;{\sin( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}} & {\cos( {\omega^{\prime}\Delta\;{\tau_{MS}( \omega^{\prime} )}} )}\end{bmatrix}.}$
 9. The system of claim 7, wherein the frequency-domainSF-ECC algorithm comprises:${{\overset{\sim}{H}}_{MS}^{- 1}( {j\;\omega} )} = \begin{bmatrix}{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} & {- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} \\{- {{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )}} & {{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )}\end{bmatrix}$${where},{{{\overset{\sim}{H}}_{IL}^{({MS})}( {j\;\omega} )} = {{\cos( {\omega{\sum\limits_{k = 1}^{M}\;{\Delta\;{t_{MS}^{(k)}(\omega)}}}} )}{\mspace{14mu}\;}{and}}}$${{\overset{\sim}{H}}_{FEXT}^{({MS})}( {j\;\omega} )} = {{- j}\;{\sin( {\omega{\sum\limits_{k = 1}^{M}\;{\Delta\;{t_{MS}^{(k)}(\omega)}}}} )}}$and the compensated digital signal vector is:${\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {{IDTFT}{\{ {{\overset{\sim}{H}}_{MS}^{- 1}{( {j\;\omega} ) \cdot {DTFT}}\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{t = \frac{n \cdot T_{s}}{SPS}}} \}} \}.}}$10. The system of claim 7, wherein the time-domain SF-ECC algorithmcomprises:$\mspace{20mu}{{{\overset{\sim}{h}}_{MS}^{- 1}(t)} = \begin{bmatrix}{{\overset{\sim}{h}}_{IL}^{({MS})}(t)} & {- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} \\{- {{\overset{\sim}{h}}_{FEXT}^{({MS})}(t)}} & {{\overset{\sim}{h}}_{IL}^{({MS})}(t)}\end{bmatrix}}$ ⁢ where , ⁢ ⁢ h ~ IL ( MS ) ⁡ ( t ) = ⁢ - 1 ⁢ { H ~ IL ( MS ) ⁡( j ⁢ ⁢ ω ) } = ⁢ ∑ k = 1 M ⁢ ⁢ h ~ IL ( MS ) ⁢ ( k ) ⁡ ( t ) , ⁢ ⁢ h ~ FEXT ( MS) ⁡ ( t ) = ⁢ - 1 ⁢ { H ~ FEXT ( MS ) ⁡ ( j ⁢ ⁢ ω ) } = ⁢ ∑ k = 1 M ⁢ ⁢ h ~ FEXT( MS ) ⁢ ( k ) ⁡ ( t ) , ⁢ h ~ IL ( MS ) ⁢ ( k ) ⁡ ( t ) = ⁢ - 1 ⁢ { H ~ IL (MS ) ⁢ ( k ) ⁡ ( j ⁢ ⁢ ω ) } = ⁢ 1 2 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ cos ⁡ ( ω ⁢ ⁢ Δ ⁢ ⁢ τ MS ⁡ ( ω )) ⁢ Π ⁡ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k ) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω ≅ ⁢ 1 2 ⁢ ⁢ π ⁢ ∫ - ∞ ∞⁢cos ⁡ ( ω ⁢ ⁢ Δ ⁢ ⁢ τ MS ( I ) ( k ) ⁡ ( ω ) ) ⁢ Π ⁡ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k ) ) ⁢e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢ 1 4 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ e j ⁢ ⁢ ω ⁡ ( Δ ⁢ ⁢ τ MS 0 ( k ) - ω0 ( k ) ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ) + j ⁢ ⁢ ω 2 ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ⁢ Π ⁢ ( j ⁢ ⁢ ω - j ⁢⁢ω 0 ( k ) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω + ⁢ 1 4 ⁢ ⁢ π ⁢ ∫ - ∞ ∞ ⁢ e - j ⁢ ⁢ ω ⁡ ( Δ ⁢ ⁢ τMS 0 ( k ) - ω 0 ( k ) ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ) + j ⁢ ⁢ ω 2 ⁢ Δ ⁢ ⁢ τ MS 1 ( k ) ⁢Π ⁢ ( j ⁢ ⁢ ω - j ⁢ ⁢ ω 0 ( k ) ) ⁢ e j ⁢ ⁢ ω ⁢ ⁢ t ⁢ ⁢ d ⁢ ⁢ ω = ⁢ A · [ j ⁢ e - j ⁢ ⁢ ϑ( k ) ⁡ ( t ) * δ ⁡ ( t + γ ( k ) ) + - j ⁢ e j ⁢ ⁢ ϑ ( k ) ⁡ ( t ) * δ ⁡ ( t -γ ( k ) ) ] * sin ⁢ ⁢ c ⁡ ( B M 2 ⁢ t ) ⁢ e j ⁢ ⁢ ω 0 ( k ) ⁢ t ⁢   and$\mspace{20mu}{{{\overset{\sim}{h}}_{FEXT}^{{({MS})}{(k)}}(t)} \cong {{A \cdot \begin{bmatrix}{{\sqrt{j}e^{{- j}\;{\vartheta^{(k)}{(t)}}}*{\delta( {t + \gamma^{(k)}} )}} +} \\{\sqrt{- j}e^{j\;{\vartheta^{(k)}{(t)}}}*{\delta( {t - \gamma^{(k)}} )}}\end{bmatrix}}*\sin\;{c( {\frac{B_{M}}{2}t} )}e^{j\;\omega_{0}^{(k)}t}}}$and the compensated digital signal vector is:${\underset{\_}{v}}_{n}^{({{SF} - {ECC}})} = {\{ {{{\overset{\sim}{h}}_{MS}^{- 1}(t)}*\{ \lbrack {{h_{MS}(t)}*{v_{in}( {t,0} )}} \rbrack_{{t = \frac{n \cdot T_{s}}{SPS}}} \}} \}.}$11. The system of claim 7, wherein the EECC algorithm can be utilizedfor coupled multiple transmission lines by using the exact inversematrix sampled coupling transfer function of the coupled multipletransmission lines.
 12. The system of claim 7, wherein the SF-ECCalgorithm can be utilized for coupled multiple transmission lines byusing the approximated frequency-domain transfer function of the coupledmultiple transmission lines or by using the approximated time-domainimpulse response of coupled multiple transmission lines.
 13. The systemof claim 12, wherein one DSP unit is located at the transmitter side ofthe optoelectronic system for pre-compensation implementation of thecoupling impairments by the EECC or SF-ECC algorithms and a second DSPunit comprising an interpolator and a FFE-DFE is located at the receiverside of the optoelectronic system.
 14. The system of claim 7, whereinthe DSP unit is extended to compensate for intersymbol interference(ISI) by addition of an interpolator and a feed-forward equalizer anddecision feedback equalizer (FFE-DFE).